In a medium saucepan, heat the soy sauce, ½ cup water, sugar, scallions, garlic, sake, dried shiitakes, and ginger. When the mixture reaches a boil, use a strainer to remove the vegetables. In a small bowl, combine the cornstarch with 1 tablespoon water to make a paste. Whisk into the sauce and bring to a simmer for 20 minutes, stirring occasionally.
Coat the chicken thighs in half of the teriyaki sauce, reserving the remaining sauce for serving. Marinate for 4-12 hours.
Preheat the oven to 400°F.
Remove the chicken from the marinade, discard marinade and season chicken with salt and pepper. Roast on a baking sheet until cooked through, about 22 minutes. Let the chicken rest for 5-10 minutes before serving.
Place the rice, 2 cups of water and a pinch of salt in a small saucepan. Bring to a boil, reduce the heat to a simmer, cover and cook until the grains are tender, 20-25 minutes.
In a blender, combine the scallions, vinegar, sugar, salt, and pepper in a blender on high until the scallions are completely puréed. With the motor running, drizzle in the canola and sesame oils until emulsified. Season to taste.
In a large bowl, toss both cabbages with the Asian dressing.
To serve, divide the rice between bowls. Top with a chicken breast and cabbage slaw. Drizzle with reserved teriyaki sauce and sprinkle with peanuts.